This study analyses the main characteristics of the fully developed laminar pulsatile flow in a toroidal pipe as the governing parameters vary. A novel computational technique is developed to obtain time-periodic solutions of the Navier--Stokes equations. They are computed as fixed points of the system in the frequency domain via the Newton-Raphson method. Drawbacks and advantages of the adopted methodology with respect to a time-stepping technique are discussed. The unsteady component of the driving pressure gradient is found to change linearly with the pulsation amplitude, with a proportionality coefficient dependent on the pulsation frequency. Although the time-averaged streamwise wall shear stress is very close to the value in the steady case, very large fluctuations are observed within the period. Flow reversal occurs during certain time intervals in the period for high pulsation amplitudes. The analysis of the spatial structure of the unsteady component of the velocity field shows that three different flow regimes can be identified, depending on the pulsation frequency, termed quasi-steady, intermediate and plug-flow regimes.

The linear temporal stability of the fully-developed pulsatile flow in a torus with high curvature is investigated using Floquet theory. The baseflow is computed via a Newton--Raphson  iteration in frequency space to obtain basic states at supercritical Reynolds numbers in the steady state for two curvatures,= 0.1 and = 0.3, exhibiting structurally different linear instabilities in the steady case. The addition of a pulsatile component is found to be overall stabilising over a wide range of pulsation amplitudes, in particular for the higher values of the curvature. The pulsatile flows are found to be at most transiently stable with large intracyclic growth rate variations even at small pulsation amplitudes. While these growth rates are likely insufficient to trigger abrupt transition at the parameters in this work, the trends indicate that this is indeed likely for higher pulsation amplitudes, similar to pulsatile flow in straight pipes. At the edge of the considered parameter range, subharmonic eigenvalue orbits in the local spectrum of the time-periodic operator, recently found in pulsating channel flow, have been confirmed also for pulsatile flow in toroidal pipes underlining the generality of this phenomenon.

Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr-Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien-Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.

Spectral degeneracies where eigenvalues and eigenvectors simultaneously coalesce, also known as exceptional points, are a natural consequence of the strong non-normality of the Orr-Sommerfeld operator describing the evolution of infinitesimal disturbances in parallel shear flows. While the resonances associated with these points give rise to algebraic growth, the development of non-modal stability theory exploiting specific perturbation structures with much larger potential for transient energy growth has led to waning interest in spectral degeneracies. The appearance of subharmonic eigenvalue orbits, recently discovered in the periodic spectrum of pulsating Poiseuille flow, can be traced back to the coalescence of eigenvalues at exceptional points. We present a thorough analysis of the spectral properties of the linear operator to identify exceptional points and accurately map the prevalence of subharmonic eigenvalue orbits for a large range of pulsation amplitudes and frequencies. This information is then combined with solutions of the linear initial value problem to analyse the impact of the appearance of these orbits on the temporal evolution of linear disturbances in pulsating Poiseuille flow. The periodic amplification phases are shown to be heralded by repeated non-normal growth bursts that are intensified by the formation of subharmonic orbits involving the leading eigenvalues. These bursts are associated with the change of alignment of the perturbation from the decaying towards the amplified branch of the subharmonic eigenvalue orbits in a so-called branch transition process.

A transient linear stability analysis of the time-dependent flow around a natural laminar flow airfoil undergoing small-amplitude pitching motion is performed using the Optimally Time-Dependent (OTD) framework. The most salient linear instabilities associated with the instantaneous basic state are computed and tracked over time. The resulting OTD modes reflect the variations in the basic state and can be used as predictors of its evolution including the formation of a laminar separation bubble, its gradual spanwise modulation via primary global instability leading to secondary instability and finally rapid breakdown to turbulence. The study confirms and expands upon earlier stability analyses of the same case based on the local properties of span-wise averaged velocity profiles in the bubble that predicted the onset of absolute instability soon followed by rapid breakdown of the separation bubble. The three-dimensional structure of the most unstable OTD mode is extracted which compares well with both the locally absolutely unstable mode and the evolution of the basic state itself.

Thin airfoil dynamic stall at moderate Reynolds numbers is typically linked to the sudden bursting of a small laminar separation bubble close to the leading edge. Given the strong sensitivity of laminar separation bubbles to external disturbances, the onset of dynamic stall on a NACA0009 airfoil section subject to different levels of low-amplitude freestream disturbances is investigated using direct numerical simulations. For turbulence intensities at the leading edge of Tu = 0.02 %, the flow is practically indistinguishable from clean inflow simulations in literature. For Tu = 0.05 %, the bursting process is found to be considerably less smooth and strong coherent vortex shedding from the laminar separation bubble is observed prior to the formation of the dynamic stall vortex. The non-linear simulations are complemented with a transient linear stability analysis of the time-dependent evolution of the laminar shear layer in the bursting separation bubble using a spatially localised formulation of the Optimally Time-Dependent (OTD) framework, with which the most unstable part of the instantaneous tangent space of the non-linear trajectory is tracked over time. The resulting modes reveal intermittent switching between two regimes. Rapid growth of the Kelvin--Helmholtz rolls on the separated shear layer and complicated secondary instabilities on the transitional part of the separation bubble. The appearance of the latter is linked to large spikes in the instantaneous growth rate within the linear subspace and more rapid transition in the non-linear baseflow. These intense growth spikes are well correlated with the subsequent shedding of energetic vortices from the laminar separation bubble.

The laminar separation bubble (LSB) forming on a symmetric NACA0009 airfoil at a steady pitch angle of 8 degrees and Reynolds number Re = 2.0 · 10^5 under theinfluence of free-stream disturbances is computed using high-fidelity large-eddy simulations. Two disturbance levels are considered corresponding to cruiseconditions achievable in low free-stream turbulence wind tunnels and higher disturbance levels typical of larger, conventional wind tunnels. The extracted flow fields are Fourier transformed in the spanwise direction and then eachspanwise mode is analysed using Spectral Proper Orthogonal Decomposition to extract the dominant structures in the flow exhibiting spatio-temporal coherence. Since LSBs are known to be sensitive to external disturbances as well as self-excited absolute instabilities, the analysis of the effect of free-stream disturbances on the separation bubble is crucial for understanding the bubble dynamics and to relate results from simulations to experiments in applications where certain levels of inflow disturbance are unavoidable.

The increased availability of large scale computing hardware brings the analysis of fully three-dimensional non-autonomous flow cases within reach. In these flow scenarios, the simplifying assumption of temporal homogeneity is not applicable and with it many data-driven analysis techniques that rely on it. Within the well-established modal decomposition framework of Proper Orthogonal Decomposition (POD), we can treat time in the same way as the spatial dimensions and apply the method to statistical ensembles of non-autonomous flows in order to extract coherent structures in space and time from the resulting experimental or numerical data, leading to the space-time POD formulation. This extension of the existing method is demonstrated on the model problem of the complex Ginzburg--Landau equation, modified to include non-autonomous parameter variations. Subsequently, the space-time POD analysis is carried out on a numerical dataset of 25 realisations of the onset of leading edge dynamic stall on a NACA0009 airfoil section subject to low levels of background disturbances. The space-time POD, combined with extended POD, is used to extract the spatio-temporal structure of energetic wavetrains during the bursting of the laminar separation bubble close to the leading edge, which are found to be statistically relevant phenomena in the context of incipient dynamic stall. The potential of the space-time POD methodology to objectively extract coherent structures from ensembles of non-autonomous data is demonstrated.